scholarly journals Natural Frequency Characteristics of the Beam with Different Cross Sections Considering the Shear Deformation Induced Rotary Inertia

2020 ◽  
Vol 10 (15) ◽  
pp. 5245
Author(s):  
Chunfeng Wan ◽  
Huachen Jiang ◽  
Liyu Xie ◽  
Caiqian Yang ◽  
Youliang Ding ◽  
...  
Keyword(s):  
Shear Deformation ◽  
Cross Sections ◽  
Timoshenko Beam ◽  
High Frequency ◽  
Natural Frequencies ◽  
Beam Theory ◽  
Equation Of Motion ◽  
Timoshenko Beam Theory ◽  
Rotary Inertia ◽  
Cross Sectional

Based on the classical Timoshenko beam theory, the rotary inertia caused by shear deformation is further considered and then the equation of motion of the Timoshenko beam theory is modified. The dynamic characteristics of this new model, named the modified Timoshenko beam, have been discussed, and the distortion of natural frequencies of Timoshenko beam is improved, especially at high-frequency bands. The effects of different cross-sectional types on natural frequencies of the modified Timoshenko beam are studied, and corresponding simulations have been conducted. The results demonstrate that the modified Timoshenko beam can successfully be applied to all beams of three given cross sections, i.e., rectangular, rectangular hollow, and circular cross sections, subjected to different boundary conditions. The consequence verifies the validity and necessity of the modification.

scholarly journals Who developed the so-called Timoshenko beam theory?

2019 ◽  
Vol 25 (1) ◽  
pp. 97-116 ◽  
Author(s):  
Isaac Elishakoff
Keyword(s):  
Shear Deformation ◽  
Correction Factor ◽  
Timoshenko Beam ◽  
World Literature ◽  
Beam Theory ◽  
Scientific Work ◽  
Timoshenko Beam Theory ◽  
Rotary Inertia ◽  
The World ◽  
First Time

The use of the Google Scholar produces about 78,000 hits on the term “Timoshenko beam.” The question of priority is of great importance for this celebrated theory. For the first time in the world literature, this study is devoted to the question of priority. It is that Stephen Prokofievich Timoshenko had a co-author, Paul Ehrenfest. It so happened that the scientific work of Timoshenko dealing with the effect of rotary inertia and shear deformation does not carry the name of Ehrenfest as the co-author. In his 2002 book, Grigolyuk concluded that the theory belonged to both Timoshenko and Ehrenfest. This work confirms Grigolyuk’s discovery, in his little known biographic book about Timoshenko, and provides details, including the newly discovered letter of Timoshenko to Ehrenfest, which is published here for the first time over a century after it was sent. This paper establishes that the beam theory that incorporates both the rotary inertia and shear deformation as is known presently, with shear correction factor included, should be referred to as the Timoshenko-Ehrenfest beam theory.

Celebrating the Centenary of Timoshenko's Study of Effects of Shear Deformation and Rotary Inertia

2015 ◽  
Vol 67 (6) ◽  
Author(s):  
Isaac Elishakoff ◽  
Julius Kaplunov ◽  
Evgeniya Nolde
Keyword(s):  
Differential Equation ◽  
Shear Deformation ◽  
Timoshenko Beam ◽  
Initial Conditions ◽  
Beam Theory ◽  
Timoshenko Beam Theory ◽  
Rotary Inertia ◽  
Governing Differential Equation ◽  
Second Spectrum

This study revisits Timoshenko beam theory (TBT). It discusses at depth a more consistent and simpler governing differential equation. The so-called second spectrum is also addressed. Then, we provide the asymptotic justification of the aforementioned differential equation along with detailed discussion of the boundary and initial conditions. The paper also presents remarks of historical character, in the context of other pertinent studies.

ANALYSIS OF AUXETIC BEAMS AS RESONANT FREQUENCY BIOSENSORS

2012 ◽  
Vol 12 (05) ◽  
pp. 1240027 ◽  
Author(s):  
TEIK-CHENG LIM
Keyword(s):  
Shear Deformation ◽  
Cross Sections ◽  
Timoshenko Beam ◽  
Poisson’S Ratio ◽  
Beam Theory ◽  
Timoshenko Beam Theory ◽  
Poisson's Ratio ◽  
Bernoulli Beam ◽  
Bernoulli Beam Theory ◽  
Euler Bernoulli Beam

The mechanics of beam vibration is of fundamental importance in understanding the shift of resonant frequency of microcantilever and nanocantilever sensors. Unlike the simpler Euler–Bernoulli beam theory, the Timoshenko beam theory takes into consideration rotational inertia and shear deformation. For the case of microcantilevers and nanocantilevers, the minute size, and hence low mass, means that the topmost deviation from the Euler–Bernoulli beam theory to be expected is shear deformation. This paper considers the extent of shear deformation for varying Poisson's ratio of the beam material, with special emphasis on solids with negative Poisson's ratio, which are also known as auxetic materials. Here, it is shown that the Timoshenko beam theory approaches the Euler–Bernoulli beam theory if the beams are of solid cross-sections and the beam material possess high auxeticity. However, the Timoshenko beam theory is significantly different from the Euler–Bernoulli beam theory for beams in the form of thin-walled tubes regardless of the beam material's Poisson's ratio. It is herein proposed that calculations on beam vibration can be greatly simplified for highly auxetic beams with solid cross-sections due to the small shear correction term in the Timoshenko beam deflection equation.

Medium Frequency Vibration Analysis of Beam Structures Modeled by the Timoshenko Beam Theory

2020 ◽  
Author(s):  
Yichi Zhang ◽  
Bingen Yang
Keyword(s):  
Shear Deformation ◽  
Timoshenko Beam ◽  
Vibration Analysis ◽  
Beam Theory ◽  
Timoshenko Beam Theory ◽  
Bernoulli Beam ◽  
Beam Structure ◽  
Medium Frequency ◽  
Beam Structures ◽  
Euler Bernoulli Beam

Abstract Vibration analysis of complex structures at medium frequencies plays an important role in automotive engineering. Flexible beam structures modeled by the classical Euler-Bernoulli beam theory have been widely used in many engineering problems. A kinematic hypothesis in the Euler-Bernoulli beam theory is that plane sections of a beam normal to its neutral axis remain normal when the beam experiences bending deformation, which neglects the shear deformation of the beam. However, as observed by researchers, the shear deformation of a beam component becomes noticeable in high-frequency vibrations. In this sense, the Timoshenko beam theory, which describes both bending deformation and shear deformation, may be more suitable for medium-frequency vibration analysis of beam structures. This paper presents an analytical method for medium-frequency vibration analysis of beam structures, with components modeled by the Timoshenko beam theory. The proposed method is developed based on the augmented Distributed Transfer Function Method (DTFM), which has been shown to be useful in various vibration problems. The proposed method models a Timoshenko beam structure by a spatial state-space formulation in the s-domain, without any discretization. With the state-space formulation, the frequency response of a beam structure, in any frequency region (from low to very high frequencies), can be obtained in an exact and analytical form. One advantage of the proposed method is that the local information of a beam structure, such as displacements, bending moment and shear force at any location, can be directly obtained from the space-state formulation, which otherwise would be very difficult with energy-based methods. The medium-frequency analysis by the augmented DTFM is validated with the FEA in numerical examples, where the efficiency and accuracy of the proposed method is present. Also, the effects of shear deformation on the dynamic behaviors of a beam structure at medium frequencies are illustrated through comparison of the Timoshenko beam theory and the Euler-Bernoulli beam theory.

On the Quest for the Best Timoshenko Shear Coefficient

2003 ◽  
Vol 70 (1) ◽  
pp. 154-157 ◽  
Author(s):  
M. B. Rubin
Keyword(s):  
Correction Factor ◽  
Timoshenko Beam ◽  
Theoretical Basis ◽  
Natural Frequencies ◽  
Beam Theory ◽  
Vibrational Frequencies ◽  
Timoshenko Beam Theory ◽  
Cosserat Theory ◽  
Shear Correction Factor ◽  
Shear Coefficient

Classical Timoshenko beam theory includes a shear correction factor κ which is often used to match natural vibrational frequencies of the beam. In this note, a number of static and dynamic examples are considered which provide a theoretical basis for specifying κ=1. Within the context of Cosserat theory, natural frequencies of the beam can be matched by appropriate specification of the director inertia coefficients with κ=1.

Application of Timoshenko Beam Theory to the Dynamics of Flexible Legged Locomotion

1988 ◽  
Vol 110 (1) ◽  
pp. 28-34
Author(s):  
E. M. Bakr ◽  
A. A. Shabana
Keyword(s):  
Shear Deformation ◽  
Timoshenko Beam ◽  
Shape Function ◽  
Reference Conditions ◽  
Beam Theory ◽  
Legged Locomotion ◽  
Rotary Inertia ◽  
Stiffness Matrices ◽  
Reference Motion ◽  
Invariant Matrices

A method for the dynamic analysis of flexible legged locomotion systems that accounts for the rotary inertia and shear deformation effects is presented. The motion of the flexible components in the legged vehicle is described using a set of inertia-variant Timoshenko beams that undergo large rotations. A shape function that accounts for the combined effect of rotary inertia and shear is employed to describe the deformation relative to a selected component reference and the rigid-body modes of the shape function are eliminated using a set of reference conditions. Kinetic and strain energies are derived for each Timoshenko beam, thus identifying the beam mass and stiffness matrices which account for the rotary inertia and shear deformation effects. A new set of time-invariant matrices that describe the nonlinear inertia coupling between the reference motion and elastic deformation and account for the rotary inertia and shear is developed and it is shown that the form of these matrices as well as the mass and stiffness matrices are significantly affected by the inclusion of rotary inertia and shear. Numerical experimentations indicate that shear and rotary inertia can have a significant effect on the dynamics of flexible legged locomotion.

Free vibration analysis of nonlocal nanobeams: a comparison of the one-dimensional nonlocal integral Timoshenko beam theory with the two-dimensional nonlocal integral elasticity theory

2021 ◽  
pp. 108128652110312
Author(s):  
Hooman Danesh ◽  
Mahdi Javanbakht
Keyword(s):  
Free Vibration ◽  
Elasticity Theory ◽  
Timoshenko Beam ◽  
Natural Frequencies ◽  
Nonlocal Parameter ◽  
Free Vibration Analysis ◽  
Beam Theory ◽  
Timoshenko Beam Theory ◽  
Two Phase ◽  
Softening Effect

Beam theories such as the Timoshenko beam theory are in agreement with the elasticity theory. However, due to the different nonlocal averaging processes, they are expected to yield different results in their nonlocal forms. In the present work, the free vibration behavior of nonlocal nanobeams is studied using the nonlocal integral Timoshenko beam theory (NITBT) and two-dimensional nonlocal integral elasticity theory (2D-NIET) with different kernels and their results are compared. A new kernel, termed the compensated two-phase (CTP) kernel, is introduced, which entirely compensates for the boundary effects and does not suffer from the ill-posedness of previous kernels. Using the finite element method, the free vibration analysis is performed for different boundary conditions based on the first three natural frequencies. For both the NITBT and 2D-NIET with both the two-phase (TP) and CTP kernels, the nonlocal parameter has a softening effect on the natural frequencies for all the boundary conditions, without observing the paradoxical behaviors of the nonlocal differential theory. For both theories, the softening effect of the nonlocal parameter is more pronounced for the TP kernel compared to the CTP kernel. The sensitivity of the 2D-NIET to the nonlocal parameter is found to be higher than that of the NITBT. Also, the softening effects for different vibration modes are compared to each other for both theories and both kernels. The obtained results can be extended for various important beam problems with nonlocal effects and help obtain a better understanding of applicable nonlocal theories.

scholarly journals Asymmetric Oscillations of AFG Microscale Nonuniform Deformable Timoshenko Beams

Vibration ◽  
2019 ◽  
Vol 2 (2) ◽  
pp. 201-221 ◽  
Author(s):  
Mergen H. Ghayesh ◽  
Ali Farajpour ◽  
Hamed Farokhi
Keyword(s):  
Timoshenko Beam ◽  
Couple Stress Theory ◽  
Beam Theory ◽  
Modified Couple Stress Theory ◽  
Functionally Graded ◽  
Timoshenko Beam Theory ◽  
Cross Sectional ◽  
Stress Theory ◽  
Size Dependent ◽  
Modified Couple Stress

A nonlinear vibration analysis is conducted on the mechanical behavior of axially functionally graded (AFG) microscale Timoshenko nonuniform beams. Asymmetry is due to both the nonuniform material mixture and geometric nonuniformity. Using the Timoshenko beam theory, the continuous models for translation/rotation are developed via an energy balance. Size-dependence is incorporated via the modified couple stress theory and the rotation via the Timoshenko beam theory. Galerkin’s method of discretization is applied and numerical simulations are conducted for a size-dependent vibration of the AFG microscale beam. Effects of material gradient index and axial change in the cross-sectional area on the force and frequency diagrams are investigated.

Shear Coefficients for Timoshenko Beam Theory

2000 ◽  
Vol 68 (1) ◽  
pp. 87-92 ◽  
Author(s):  
J. R. Hutchinson
Keyword(s):  
Cross Sections ◽  
Timoshenko Beam ◽  
Circular Cross Section ◽  
Beam Theory ◽  
Timoshenko Beam Theory ◽  
Full Agreement ◽  
Shear Coefficient ◽  
Slender Beams ◽  
The Subject ◽  
New Formula

The Timoshenko beam theory includes the effects of shear deformation and rotary inertia on the vibrations of slender beams. The theory contains a shear coefficient which has been the subject of much previous research. In this paper a new formula for the shear coefficient is derived. For a circular cross section, the resulting shear coefficient that is derived is in full agreement with the value most authors have considered “best.” Shear coefficients for a number of different cross sections are found.

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